http://arxiv.org/abs/2110.13508
This is the Part I paper of our series of papers on a gauge-invariant perturbation theory on the Schwarzschild background spacetime. After reviewing our general framework of the gauge-invariant perturbation theory, we propose the strategy of the gauge-invariant perturbation theory on the Schwarzschild spacetime. In the above general framework, the “zero-mode problem” was a remaining important problem to develop gauge-invariant perturbation theories on generic background spacetime. In perturbation theories on the Schwarzschild background spacetime, $l=0,1$-mode problem corresponds to the above “zero-mode problem.” The above strategy proposed in this paper is a resolution of this $l=0,1$-mode problem in perturbations on the Schwarzschild background spacetime. Following this proposal, we derive the linearized Einstein equations for any modes of $l\geq 0$ in gauge-invariant manner. We discuss the solutions to the odd-mode perturbation equations in the linearized Einstein equations and show that these perturbations include the Kerr parameter perturbation in these odd-mode perturbation, which is physically reasonable. In the Part II and Part III papers [K.~Nakamura, arXiv:21XX.XXXXX; arXiv:21XX.XXXXX.] of this series of papers, we will show that the even-mode solutions to the linearized Einstein equations obtained through our proposal are also physically reasonable. Then, we conclude that our proposal of the resolution of the $l=0,1$-mode problem is also physically reasonable.
K. Nakamura
Wed, 27 Oct 21
80/80
Comments: 52 pages, 3 figures, The Part I paper of the full paper version of the previous short papers arXiv:2102.083v3[gr-qc]; arXiv:2102.10650v3[gr-qc] (v1)